Simplify ((2y^2-5y-3)/(y-4))÷(y^2+y-12)

$\frac{2 y^{2} - 5 y - 3}{y - 4} \div (y^{2} + y - 12)$

Rewrite the division as a fraction.

$\frac{\frac{2 y^{2} - 5 y - 3}{y - 4}}{y^{2} + y - 12}$

Multiply the numerator by the reciprocal of the denominator.

$\frac{2 y^{2} - 5 y - 3}{y - 4} \cdot \frac{1}{y^{2} + y - 12}$

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 3 = - 6$ and whose sum is $b = - 5$ .

Factor $- 5$ out of $- 5 y$ .

$\frac{2 y^{2} - 5 (y) - 3}{y - 4} \cdot \frac{1}{y^{2} + y - 12}$

Rewrite $- 5$ as $1$ plus $- 6$

$\frac{2 y^{2} + (1 - 6) y - 3}{y - 4} \cdot \frac{1}{y^{2} + y - 12}$

Apply the distributive property.

$\frac{2 y^{2} + 1 y - 6 y - 3}{y - 4} \cdot \frac{1}{y^{2} + y - 12}$

Multiply $y$ by $1$ .

$\frac{2 y^{2} + y - 6 y - 3}{y - 4} \cdot \frac{1}{y^{2} + y - 12}$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$\frac{(2 y^{2} + y) - 6 y - 3}{y - 4} \cdot \frac{1}{y^{2} + y - 12}$

Factor out the greatest common factor (GCF) from each group.

$\frac{y (2 y + 1) - 3 (2 y + 1)}{y - 4} \cdot \frac{1}{y^{2} + y - 12}$

Factor the polynomial by factoring out the greatest common factor, $2 y + 1$ .

$\frac{(2 y + 1) (y - 3)}{y - 4} \cdot \frac{1}{y^{2} + y - 12}$

Factor $y^{2} + y - 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 12$ and whose sum is $1$ .

$- 3, 4$

Write the factored form using these integers.

$\frac{(2 y + 1) (y - 3)}{y - 4} \cdot \frac{1}{(y - 3) (y + 4)}$

Simplify terms.

Cancel the common factor of $y - 3$ .

Factor $y - 3$ out of $(2 y + 1) (y - 3)$ .

$\frac{(y - 3) (2 y + 1)}{y - 4} \cdot \frac{1}{(y - 3) (y + 4)}$

Cancel the common factor.

$\frac{(y - 3) (2 y + 1)}{y - 4} \cdot \frac{1}{(y - 3) (y + 4)}$

Rewrite the expression.

$\frac{2 y + 1}{y - 4} \cdot \frac{1}{y + 4}$

Multiply $\frac{2 y + 1}{y - 4}$ by $\frac{1}{y + 4}$ .

$\frac{2 y + 1}{(y - 4) (y + 4)}$

Simplify ((2y^2-5y-3)/(y-4))÷(y^2+y-12)

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